Multiple bifurcations in a predator–prey system with monotonic functional response. (English) Zbl 1102.34031

A predator-prey system is treated with Holling-type-III functional response and constant harvesting of the predator. The system is transformed into some normal form. Then, the parameters are fixed numerically except the death rate of predator and the harvesting constant. With the latter two as bifurcation parameters, performing several transformations, it is shown that the system undergoes a Bogdanov-Takens bifurcation at a certain value of the parameters.


34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34C23 Bifurcation theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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[1] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 73, 1530-1535 (1992)
[2] Bogdanov, R., Bifurcations of a limit cycle for a family of vector fields on the plan, Selecta Math. Soviet., 1, 373-388 (1981)
[3] Bogdanov, R., Versal deformations of a singular point on the plan in the case of zero eigen-values, Selecta Math. Soviet., 1, 389-421 (1981) · Zbl 0518.58030
[4] Chen, J. P.; Zhang, H. D., The qualitative analysis of two species predator-prey model with Holling’s type III functional response, Appl. Math. Mech., 7, 1, 73-80 (1986)
[5] Chow, S. N.; Hale, J. K., Methods of Bifurcation Theory (1982), Springer-Verlag: Springer-Verlag New York, Heidelberg, Berlin · Zbl 0487.47039
[6] Chow, S. N.; Li, C. Z.; Wang, D., Normal Forms and Bifurcation of Planar Vector Fields (1994), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0804.34041
[7] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0448.92023
[8] Hesaaraki, M.; Moghadas, S. M., Nonexistence of limit cycles in a predator-prey system with a sigmoid functional response, Cana. Appl. Math. Quar., 7, 4, 1-8 (1999) · Zbl 0977.92019
[9] Holling, C. S., The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Ent. Sec. Can., 45, 1-60 (1965)
[10] Ivlev, V. S., Experimental Ecology of the Feeding of Fishes (1961), Yale University Press
[11] Kazarinoff, N. D.; Van Den Driessche, P., A model predator-prey system with functional response, Math. Biosci., 39, 1-2, 125-134 (1978) · Zbl 0382.92007
[12] Wang, L. L.; Li, W. T., Existence of periodic solutions of a delayed predator-prey system with functional response, Int. J. Math. Sci., 1, 55-63 (2002) · Zbl 1075.34067
[13] Wang, L. L.; Li, W. T., Existence and global stability of positive periodic solutions of a predator-prey system with delays, Appl. Math. Comput., 146, 1, 167-185 (2003) · Zbl 1029.92025
[14] Wang, L. L.; Li, W. T., Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response, J. Comput. Appl. Math., 162, 341-357 (2004) · Zbl 1076.34085
[15] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0914.58025
[16] May, R. M., Complexity and Stability in Model Ecosystems (1973), Princeton University Press: Princeton University Press Princeton, NJ
[17] Murry, J. D., Mathematical Biology (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0682.92001
[18] Myerscough, M. R.; Darwen, M. J.; Hogarth, W. L., Stability, persistence and structural stability in a classical predator-prey model, Ecol. Model., 89, 31-42 (1996)
[19] Rosenzweig, M. L., Paradox of enrichment: destabilization of exploitation ecosystem in ecological time, Science, 171, 385-387 (1971)
[20] Ruan, S.; Xiao, D., Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61, 1445-1472 (2001) · Zbl 0986.34045
[21] Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differen. Equat., 188, 135-163 (2003) · Zbl 1028.34046
[22] Xiao, D.; Ruan, S., Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differen. Equat., 176, 494-510 (2001) · Zbl 1003.34064
[23] Xiao, D.; Ruan, S., Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fieids Institute Commun., 21, 493-506 (1999) · Zbl 0917.34029
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